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Ecclesiastes 3:21 (NASB)
Who knows that the breath of man ascends upward and the breath of the beast descends downward to the earth?

Later, though, we see that the author of Ecclesiastes believes in the place called "Sheol"

Ecclesiastes 9:10 (NASB)
Whatever your hand finds to do, do it with all your might; for there is no activity or planning or knowledge or wisdom in Sheol where you are going.

He describes this in 9:5-6:

Ecclesiastes 9:5-6 (NASB)
5 For the living know they will die; but the dead do not know anything, nor have they any longer a reward, for their memory is forgotten. 6 Indeed their love, their hate and their zeal have already perished, and they will no longer have a share in all that is done under the sun.

Sheol is often translated as "the grave", but that translation is questionable.

Professors James Tabor believe that Sheol is a place of nothingness that occurs after death, where both the righteous and unrighteous go.

One of the themes of Ecclesiastes is the ignorance of the 'Preacher' and his intended audience, especially when it comes to the future, eg in chapter 2:

18Thus I hated all the fruit of my labor for which I had labored under the sun, for I must leave it to the man who will come after me. 19And who knows whether he will be a wise man or a fool? Yet he will have control over all the fruit of my labor for which I have labored by acting wisely under the sun. This too is vanity. NASB

and chapter 11:

1Cast your bread on the surface of the waters, for you will find it after many days. 2Divide your portion to seven, or even to eight, for you do not know what misfortune may occur on the earth. 3If the clouds are full, they pour out rain upon the earth; and whether a tree falls toward the south or toward the north, wherever the tree falls, there it lies. 4He who watches the wind will not sow and he who looks at the clouds will not reap. 5Just as you do not know the path of the wind and how bones are formed in the womb of the pregnant woman, so you do not know the activity of God who makes all things.

6Sow your seed in the morning and do not be idle in the evening, for you do not know whether morning or evening sowing will succeed, or whether both of them alike will be good. NASB

And it seems clear that while the author understands that we have an eternal home (at least by chapter 12), the nature of that home is something he knows very little about:

5Furthermore, men are afraid of a high place and of terrors on the road; the almond tree blossoms, the grasshopper drags himself along, and the caperberry is ineffective. For man goes to his eternal home while mourners go about in the street. NASB

He does, however, understand that God will judge all, and concludes God is to be feared regardless. In the context of the rest of the book, which refers overwhelmingly to a lack of justice in this life, he must be referring to justice after death:

12:13The conclusion, when all has been heard, is: fear God and keep His commandments, because this applies to every person. 14For God will bring every act to judgment, everything which is hidden, whether it is good or evil. NASB